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 Perfect Shells

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nick
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PostSubject: Perfect Shells   Fri Oct 16, 2009 12:00 pm

===== 3 (+1) gangs on 3 members, two pieces =======
{1, 2, 3} optional
{ 1, 2 }
{ 1, 3 }
{ 2, 3 }


===== 7 (+1) gangs on 4 members, four pieces =======
{ 1, 2, 3, 4 } optional
{ 1, 2, 3}
{ 1, 3, 4 }
{ 1, 2, 4 }
{ 2, 3, 4 }
{ 1, 2 }
{ 1, 3 }
{ 1, 4 } xor { 2, 3 }


===== 15 (+1) gangs on 5 members, four pieces =======
{ 1, 2, 3, 4, 5 } optional
{ 1, 2, 3, 4 }
{ 1, 2, 3, 5 }
{ 1, 2, 4, 5 }
{ 1, 3, 4, 5 }
{ 2, 3, 4, 5 }
{ 1, 2, 3 }
{ 1, 2, 4 }
{ 1, 2, 5 }
{ 1, 3, 4 }
{ 1, 3, 5 }
{ 1, 4, 5 }
{ 2, 3, 4 }
{ 2, 3, 5 }
{ 2, 4, 5 }
{ 1, 2 } xor { 3, 4, 5 }


===== 21 gangs on 6 members =======
ab bc ac
ab1 bc1 ac1
ab2 bc2 ac2
ab3 bc3 ac3
ab12 bc12 ac12
ab23 bc13 ac23
ab13 bc23 ac13


Last edited by nick on Sun Oct 18, 2009 12:41 pm; edited 8 times in total
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PostSubject: Infinite sequences of perfect shells   Fri Oct 16, 2009 12:18 pm

Two infinite sequences of 15,16, 63,64, 255,256
4^N - 1 (+1, optional) gangs on
2N+1 members,example for N=6
generalization of of the above 15-shell, N>= 2

---------------------------
the whole base of 13, optional : 1
all 12-sets ...............: C(13,1) = C(12,0) + C(12,1)
all 11-sets containing a or b ..: C(12,2) + C(12,2) - C(11,2)
all 10-sets containing a or b ..: C(12,3) + C(12,3) - C(11,3)
all 9-sets containing a or b ....: C(12,4) + C(12,4) - C(11,4)
all 8-sets containing a or b ....: C(12,5) + C(12,5) - C(11,5)
all 7-sets containing a or b ....: C(12,6) + C(12,6) - C(11,6)
all 6-sets containing a and b ..: C(11,4)
all 5-sets containing a and b ..: C(11,3)
all 4-sets containing a and b ..: C(11,2)
all 3-sets containing a and b ..: C(11,1)
the set { a , b } xor its complement : 1

Two infinite sequences of 31,32, 127,128, 511,512
4^N/2 - 1 (+1, optional) gangs
on 2N members, N>= 3, example for N=7
---------------------------
the whole base of 14, optional : 1
all 13-sets ...............: C(14,1) = C(13,0) + C(13,1)
all 12-sets containing a or b ..: C(13,2) + C(13,2) - C(12,2)
all 11-sets containing a or b ..: C(13,3) + C(13,3) - C(12,3)
all 10-sets containing a or b ..: C(13,4) + C(13,4) - C(12,4)
all 9-sets containing a or b ....: C(13,5) + C(13,5) - C(12,5)
all 8-sets containing a or b ....: C(13,6) + C(13,6) - C(12,6)
all 7-sets containing a .........: C(13,6)
all 6-sets containing a and b ..: C(12,4)
all 5-sets containing a and b ..: C(12,3)
all 4-sets containing a and b ..: C(12,2)
all 3-sets containing a and b ..: C(12,1)
the set { a , b } xor its complement : 1


Last edited by nick on Sun Oct 18, 2009 1:32 pm; edited 7 times in total
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PostSubject: Infinite colections of perfect shells   Sun Oct 18, 2009 12:46 pm

Let A be a set of individuals (members). A family is a collections of sets of powerset(A) (without the emptyset);

When every two set in this family have a non-empty intersection, the family is an intersecting family, or a shell (my wording).

The sets inside a shell are gangs;

A pivot is an individual common to all sets of a family. Its family is a stared family.

The trail of a family F is another family that contains all the subsets of the sets of F; ex:
{1, 2} , {1, 3, 4 } produces the trail {1}, {2}, {3} {1,2} , {1,3}, {1,4}, {1,3,4}.

A star is a family that has a pivot and contains some of its trail, i.e. those sets in the trail that contain the pivot ex:
{1}, {1,2}, {1,3},{1,2.3}, {1,4} has the pivot 1.

A perfect shell is a shell that has the same cardinality with the stars generated by each member ex

| {1,2},{2,3},{1,3} |
= | {1},{1,2},{1,3} | = Star(1)
= | {2},{2,1},{2,3} | = Star(2)
= | {3},{3,1},{3,2} | = Star(3)

I think it is funny to search for perfect shells.

[ to be continued...]
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PostSubject: union product   Thu Oct 22, 2009 10:38 pm

Take A = powerset (1, 2, 3)
Take B = {a, b}, {a, c}, {b, c}, {a, b, c}

Take A xu B, the union product of A and B := the family of all reunions of two sets, first in A and the second in B.

{ 1, 2, 3, a, b, c } is optional, and we get a 31-32 perfect shell on 6 members, which contains the 21 previous shell. In fact we have
3x7 = 21 and (3+1)x(7+1)-1 = 31

ab bc ac abc
ab1 bc1 ac1 abc1
ab2 bc2 ac2 abc2
ab3 bc3 ac3 abc3
ab12 bc12 ac12 abc12
ab23 bc13 ac23 abc23
ab13 bc23 ac13 abc13
ab123 bc123 ac123 (abc123 is optional)



==== Until now ====
Let HP[n] denote the perfect shell built by previous method (vivat Maple)(HP means Half Power)

HP[2]* = {a}
HP[2] = {a} {a,b}
HP[3]* = {a, b}, {b, c}, {a, c}
HP[3] = {a, b}, {b, c}, {a, c}, {a, b, c}
HP[4]* = {a, b}, {a, c}, {a, d}, {a, b, c} , {a, b, d}, {a, c, d}, {b, c, d}
HP[4] = HP[4]* and {a, b, c, d}

and so on.

i) A method to build new perfect shells is to take the union product of a perfect shell with a power set. Example:
HP[3]* xu Powerset [3]* ------> the previous 21 perfect shell.

ii) Another method is by factorization, taking a = b in a shell;

iii) Also, by changing small sets in the HP series with their complementaries we get a bundle of new perfect shells. The extrema is a new series of half powers:
- for n odd, take all subsets larger than (n+1)/2
- for n even, take all subsets larger than n/2 + 1, then add half of the n/2-subsets : for each non-intersecting couple, choose one of the couple.
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